A good confidence signal \(c\)  should give rise to calibrated confidence predictions; Namely, the binary classifier \(c\circ f \in [0,1]\)  should be calibrated. However, calibration by itself does not guarantee that the confidence measure is useful. For example, the confidence measure that always output the marginal accuracy \(c(x) = \Pr [1 - (X,Y)]\)  will be perfectly calibrated but useless for practical purposes. Thus, a good confidence measure should also give rise to a variety of confidence values; this requirement is referred to as sharpness [1]}Another approach is to consider calibration on overlapping structured subgroups of the data [2]}, [3]}, but we focus instead on global calibration and sharpness.. Letting \(T(x) = \textbf {E}[(X,Y) \vert c(X) = c(x)]\) , we can decompose the \(\ell _2\)  “loss”
of a confidence measure \(c\)  as:
\(\textbf {E}[((X,Y)-c(X))^{2}] =\textbf {Var}[(X,Y)]-\underbrace{\textbf {Var}[T(x)]}_{\text{sharpness}}+\underbrace{\textbf {E}[(T(x)-c(X))^{2}]}_{\text{calibration error}}\) 
